Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(1304916060416\)\(\medspace = 2^{8} \cdot 1721^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.561440134993984.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T272 |
| Parity: | even |
| Determinant: | 1.1721.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.561440134993984.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} - 2x^{6} - 4x^{5} + 62x^{4} + 6x^{3} + x^{2} - 168x - 320 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 107 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 107 }$:
\( x^{2} + 103x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 9\cdot 107 + 32\cdot 107^{2} + 92\cdot 107^{3} + 58\cdot 107^{4} + 3\cdot 107^{5} + 78\cdot 107^{6} + 11\cdot 107^{7} + 83\cdot 107^{8} + 15\cdot 107^{9} +O(107^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 62 a + 101 + \left(62 a + 13\right)\cdot 107 + \left(8 a + 71\right)\cdot 107^{2} + \left(46 a + 15\right)\cdot 107^{3} + \left(84 a + 94\right)\cdot 107^{4} + \left(81 a + 52\right)\cdot 107^{5} + \left(68 a + 44\right)\cdot 107^{6} + \left(22 a + 54\right)\cdot 107^{7} + \left(36 a + 75\right)\cdot 107^{8} + \left(8 a + 51\right)\cdot 107^{9} +O(107^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 88 a + 94 + \left(69 a + 90\right)\cdot 107 + \left(35 a + 3\right)\cdot 107^{2} + \left(39 a + 71\right)\cdot 107^{3} + \left(38 a + 99\right)\cdot 107^{4} + \left(40 a + 23\right)\cdot 107^{5} + \left(51 a + 56\right)\cdot 107^{6} + \left(62 a + 61\right)\cdot 107^{7} + \left(43 a + 101\right)\cdot 107^{8} + \left(91 a + 54\right)\cdot 107^{9} +O(107^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 78 a + 3 + \left(64 a + 44\right)\cdot 107 + \left(12 a + 20\right)\cdot 107^{2} + \left(84 a + 27\right)\cdot 107^{3} + \left(a + 42\right)\cdot 107^{4} + \left(7 a + 8\right)\cdot 107^{5} + \left(95 a + 49\right)\cdot 107^{6} + \left(24 a + 104\right)\cdot 107^{7} + \left(2 a + 10\right)\cdot 107^{8} + \left(88 a + 37\right)\cdot 107^{9} +O(107^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 86 + 95\cdot 107 + 67\cdot 107^{2} + 21\cdot 107^{3} + 103\cdot 107^{4} + 75\cdot 107^{5} + 67\cdot 107^{6} + 71\cdot 107^{7} + 71\cdot 107^{8} + 97\cdot 107^{9} +O(107^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 19 a + 18 + \left(37 a + 68\right)\cdot 107 + \left(71 a + 76\right)\cdot 107^{2} + \left(67 a + 85\right)\cdot 107^{3} + \left(68 a + 106\right)\cdot 107^{4} + \left(66 a + 39\right)\cdot 107^{5} + \left(55 a + 7\right)\cdot 107^{6} + \left(44 a + 46\right)\cdot 107^{7} + \left(63 a + 106\right)\cdot 107^{8} + \left(15 a + 55\right)\cdot 107^{9} +O(107^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 29 a + 101 + \left(42 a + 10\right)\cdot 107 + \left(94 a + 6\right)\cdot 107^{2} + \left(22 a + 30\right)\cdot 107^{3} + \left(105 a + 72\right)\cdot 107^{4} + \left(99 a + 34\right)\cdot 107^{5} + \left(11 a + 101\right)\cdot 107^{6} + \left(82 a + 1\right)\cdot 107^{7} + \left(104 a + 102\right)\cdot 107^{8} + \left(18 a + 65\right)\cdot 107^{9} +O(107^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 45 a + 28 + \left(44 a + 95\right)\cdot 107 + \left(98 a + 42\right)\cdot 107^{2} + \left(60 a + 84\right)\cdot 107^{3} + \left(22 a + 64\right)\cdot 107^{4} + \left(25 a + 81\right)\cdot 107^{5} + \left(38 a + 23\right)\cdot 107^{6} + \left(84 a + 76\right)\cdot 107^{7} + \left(70 a + 90\right)\cdot 107^{8} + \left(98 a + 48\right)\cdot 107^{9} +O(107^{10})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(3,6)(4,7)$ | $-3$ |
| $9$ | $2$ | $(1,5)(2,8)(3,6)(4,7)$ | $1$ |
| $12$ | $2$ | $(1,2)$ | $-3$ |
| $24$ | $2$ | $(1,3)(2,4)(5,6)(7,8)$ | $3$ |
| $36$ | $2$ | $(1,2)(3,7)$ | $1$ |
| $36$ | $2$ | $(1,2)(3,6)(4,7)$ | $1$ |
| $16$ | $3$ | $(2,8,5)$ | $0$ |
| $64$ | $3$ | $(2,8,5)(4,7,6)$ | $0$ |
| $12$ | $4$ | $(3,4,6,7)$ | $3$ |
| $36$ | $4$ | $(1,2,5,8)(3,4,6,7)$ | $1$ |
| $36$ | $4$ | $(1,2,5,8)(3,6)(4,7)$ | $-1$ |
| $72$ | $4$ | $(1,3,5,6)(2,4,8,7)$ | $-1$ |
| $72$ | $4$ | $(1,2)(3,4,6,7)$ | $-1$ |
| $144$ | $4$ | $(1,3,2,7)(4,5)(6,8)$ | $-1$ |
| $48$ | $6$ | $(2,5,8)(3,6)(4,7)$ | $0$ |
| $96$ | $6$ | $(2,8,5)(3,4)$ | $0$ |
| $192$ | $6$ | $(1,3)(2,7,8,6,5,4)$ | $0$ |
| $144$ | $8$ | $(1,3,2,4,5,6,8,7)$ | $1$ |
| $96$ | $12$ | $(2,8,5)(3,4,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.